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On existence and uniqueness for transport equations with non-smooth velocity fields under inhomogeneous Dirichlet data
Description
A transport equation with a non-smooth velocity field is considered under inhomogeneous Dirichlet boundary conditions. The spatial gradient of the velocity field is assumed in Lp′in space and the divergence of the velocity field is assumed to be bounded. By introducing a suitable notion of solutions, it is shown that there exists a unique renormalized weak solution for Lp initial and boundary data for 1/p+1/p′ = 1. Our theory is considered as a natural extension of the theory due to DiPerna and Lions (1989), where there is no boundary. Although a smooth domain is considered, it is allowed to be unbounded. A key step is a mollification of a solution. In our theory, mollification in the direction normal to the boundary is tailored to approximate the boundary data.
Journal
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- Hokkaido University Preprint Series in Mathematics
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Hokkaido University Preprint Series in Mathematics 1162 1-30, 2025-01-23
Department of Mathematics, Hokkaido University
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Details 詳細情報について
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- CRID
- 1390303000764753920
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- DOI
- 10.14943/112379
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- HANDLE
- 2115/93942
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- Text Lang
- en
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- Article Type
- departmental bulletin paper
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- Data Source
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- JaLC
- IRDB
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- Abstract License Flag
- Allowed